Problem: Is ${537668}$ divisible by $9$ ?
Solution: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {537668}= &&{5}\cdot100000+ \\&&{3}\cdot10000+ \\&&{7}\cdot1000+ \\&&{6}\cdot100+ \\&&{6}\cdot10+ \\&&{8}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {537668}= &&{5}(99999+1)+ \\&&{3}(9999+1)+ \\&&{7}(999+1)+ \\&&{6}(99+1)+ \\&&{6}(9+1)+ \\&&{8} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {537668}= &&\gray{5\cdot99999}+ \\&&\gray{3\cdot9999}+ \\&&\gray{7\cdot999}+ \\&&\gray{6\cdot99}+ \\&&\gray{6\cdot9}+ \\&& {5}+{3}+{7}+{6}+{6}+{8} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${537668}$ is divisible by $9$ if ${ 5}+{3}+{7}+{6}+{6}+{8}$ is divisible by $9$ Add the digits of ${537668}$ $ {5}+{3}+{7}+{6}+{6}+{8} = {35} $ If ${35}$ is divisible by $9$ , then ${537668}$ must also be divisible by $9$ ${35}$ is not divisible by $9$, therefore ${537668}$ must not be divisible by $9$.